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Full text of "Scientific Papers - Vi"

1918]
ON  THE  THEORY  OF  LUBRICATION
527
and he gives (in a different notation) k = 2-2.    For values of k equal to 2'0,
2'1, 2'2, 2'3 I find for the coefficient of c2//^2 on the right of (15) respectively
•02648,    -02665,    -02670,    -02663.
In agreement with Keynolds the maximum occurs when k = 2-2 nearly, and the maximum value is
P = 0-1602 ^~-............................(18)*
It should be observed—=-and it is true whatever value be taken for k—that P varies as the square of cjh^.
With the above value of k, viz. 2-2,
# = 1-37/1!,    ..............................(19)
fixing the place of maximum pressure.
Again, from (16) with the same value of k,
x-a = 0-4221 c,    ...........................(20)
which gives the distance of the centre of pressure from the trailing edge. And, again with the same value of k, by (17)
JfyP = 4-70 k/c............................(21)
Since /?j may be very small, it would seem  that F may be reduced to insignificance f.
In (18)—(21) the choice of k has been such as to make P a maximum. An alternative would be to make FjP a minimum. But it does not appear that this would make much practical difference. In Michell's bearings it is the position of the centre of pressure which determines the value of k by (16). If we use (20), k will be 2'2, or thereabouts, as above.
When in (16) k is very large, the right-hand member tends to zero, as also does a/c, so that x — a tends to vanish, c being given. As might be expected, the centre of pressure is then close to the trailing edge. On the other hand, when k exceeds unity but little, the right-hand member of (16) assumes an indeterminate form. When we evaluate it, we find
x — a, = %c.
For all values of k (> 1) the centre of pressure lies nearer the narrower end of the layer of fluid.
The above calculations suppose that the second surface is plane. The question suggests itself whether any advantage would arise from another choice of form. The integrations are scarcely more complicated if we take
7                 tn                                                                             /OO \
71 = mx..................................(**)
[* It may be proved that P has only one maximum when k > 1. t Although the ratio FjP diminishes with hi, F itself increases as 1/ftj.   By (18) and (21) we
have F= -75   ,— , when fe=2-2. Ill
W. F. S.]l factor is e~ikt>. As in the ordinary theory of specular reflection, the same is true for every point in the plane AOA and therefore for the element of surface at A A whose volume is 27rRdRd£. For points in a plane